Main analysis
Starting out with a prior predictive simulation illustrating why wide priors are not all that good.
set.seed(123)
m1 <- ulam(
alist(
B83bc_dik ~ dbinom(1, p),
logit(p) <- a[Bscen] + b[Bscen]*cp1,
a[Bscen] ~ dnorm(0,2) , # These intercepts are wide in regards to the probability in the data.
b[Bscen] ~ dnorm (0,2)
),
data = T1v , chains=4, cores = 2, sample = TRUE, log_lik = TRUE)Extracting priors and converting parameters to the outcome scale with the use of the inverse-link function. The dens plot below illustrates two peaks at 0 and 1, indicating the model expects, before seeing the data, that most individuals are either mostly unlikely (0) or likely (1) of intending theft. A narrower prior used further down (0, 1.5) indicates instead that a the probability is more evenly spread out and as such seen as a more “weak” prior.
pa.m1 <- inv_logit(prior.m1$a) #Intercept
pb.m1 <- inv_logit(prior.m1$b) #Slope 
A similar exercise as above, but with smaller priors ( ~ dnorm(0, 1.5)) is presented below. The dens-plot illustrates the prior predictive simulation for the use of somewhat narrower priors (i.e. ~dnorm(0, 1.5)).
Running with these priors, what follows are point estimates for the projected likelihood of theft at time 1 and time 2 controlling for individual crime propensity, followed by the projected likelihood of violence at time 1 and time 2.
Main analysis, theft, time 1.
The plot below illustrates how crime propensity is associated with the projected likelihood of theft. It is clear that different scenarios entail different intercepts (X = 0 = mean of crime propensity), but that the slopes do not differ substantially: they are all positive (i.e. the higher the crime propensity, the higher the likelihood of projected theft)
set.seed(1995)
post.mst1 <- extract.samples(mst1)
a_post <- inv_logit(post.mst1$a)
b_post <- inv_logit(post.mst1$b)
plot(precis(as.data.frame(a_post)), xlim=c(0,1), labels=Bscen_stold, main ="Intercepts, theft, Time 1") 
plot(precis(as.data.frame(b_post)), xlim = c(0,1), labels=Bscen_stold, main = "Slopes, theft, time 1")
Below you find the point estimates for each scenario on a logit scale. As a reminder which scenario is which:
| What | Scenario | Motivation | Monitoring |
|---|---|---|---|
| Theft, T1 | |||
| 1: | 100 SEK (≈ 10 €) | None | |
| 2: | 100 SEK | Police officer | |
| 3: | 2000 SEK (≈ 200 €) | None | |
| 4: | 2000 SEK | Police officer |
## mean sd 5.5% 94.5% n_eff Rhat
## a[1] 2.32 0.31 1.84 2.84 1979.50 1
## a[2] -0.97 0.20 -1.28 -0.64 2570.19 1
## a[3] 1.03 0.22 0.69 1.38 2558.79 1
## a[4] -0.87 0.21 -1.22 -0.53 2715.22 1
## b[1] 1.16 0.63 0.15 2.22 2336.00 1
## b[2] 1.13 0.41 0.47 1.82 2530.96 1
## b[3] 1.08 0.41 0.44 1.73 2042.67 1
## b[4] 1.21 0.38 0.59 1.81 2087.38 1Main analysis, theft, time 2.
Similar as above but instead projected theft at time 2.
set.seed(374)
post.mst2 <- extract.samples(mst2)
a_post.mst2 <- inv_logit(post.mst2$a)
b_post.mst2 <- inv_logit(post.mst2$b)
plot(precis(as.data.frame(a_post.mst2)), xlim=c(0,1), labels=Cscen_stold, main = "Intercepts, theft, time 2")
plot(precis(as.data.frame(b_post.mst2)), xlim=c(0,1), labels=Cscen_stold, main = "Slopes, theft, time 2")
And the point estimates on a logit scale, as well as a reminder which scenario is which:
| What | Scenario | Motivation | Monitoring |
|---|---|---|---|
| : Theft, T2 | |||
| 1: | 100 SEK (≈ 10 €) | None | |
| 2: | 100 SEK | Security guard | |
| 3: | 1000 SEK (≈ 100 €) | None | |
| 4: | 1000 SEK | Security guard |
## mean sd 5.5% 94.5% n_eff Rhat
## a[1] -0.64 0.18 -0.93 -0.36 2473.57 1
## a[2] -1.74 0.25 -2.15 -1.36 2223.88 1
## a[3] -0.79 0.19 -1.09 -0.49 2389.78 1
## a[4] -1.63 0.29 -2.10 -1.19 2192.18 1
## b[1] 1.47 0.40 0.84 2.13 2396.20 1
## b[2] 1.17 0.55 0.29 2.04 2155.07 1
## b[3] 1.65 0.46 0.96 2.42 2234.88 1
## b[4] 2.20 0.66 1.16 3.24 1883.83 1Comparison: theft, T1 vs T2.
The results indicate that adolescents would, in general, be more likely to keep the money under scenarios without monitoring (regardless of motivation) at both time 1 and time 2. The difference is more pronounced at T1 than T2. Scenario 1 (100:-/no one) differs reliably from scenario 3 (2000:-/no one) at T1 comparing the 89% credibility intervals (see point estimates). That is, adolescents were more likely to keep 100:- than 2000:- in the absence of monitoring. The same is not true at T2. At time 2 instead, adolescents responding to the scenarios without monitoring were more likely to keep the money than adolescents responding to scenarios with monitoring. But, level of motivation was irrelevant.
Across scenarios, more crime prone adolescents were more likely to keep the money. The tendency of a steeper slope for the most crime prone adolscents were less sensitive to deterrence (monitoring).
All in all, the results do speak in favor of the theoretical workings of the perception-choice process since level of monitoring affects “choice”-part.
Main analysis, violence, time 1.
post.vt1 <- extract.samples(vt1)
a_post.vt1 <- inv_logit(post.vt1$a)
b_post.vt1 <- inv_logit(post.vt1$b)
plot(precis(as.data.frame(a_post.vt1)), xlim=c(0,1), labels=Bscen_lab_v, main = "Intercepts, violence, time 1") 
plot(precis(as.data.frame(b_post.vt1)), xlim = c(0,1), labels=Bscen_lab_v, main = "Slopes, violence, time 2") 
## mean sd 5.5% 94.5% n_eff Rhat
## a[1] -1.31 0.22 -1.67 -0.97 2056.12 1
## a[2] -2.31 0.33 -2.83 -1.80 2036.88 1
## a[3] -1.13 0.23 -1.51 -0.77 2298.58 1
## a[4] -1.63 0.25 -2.03 -1.24 1848.60 1
## b[1] 0.90 0.44 0.21 1.62 2151.90 1
## b[2] 1.67 0.55 0.81 2.57 2271.04 1
## b[3] 2.07 0.48 1.34 2.86 2522.48 1
## b[4] 0.96 0.44 0.26 1.68 1664.02 1Main analysis, violence, time 2.
A plot over intercepts and slopes for projected violence at time 2.
As a reminder what the different scenarios entail:
| What | Scenario | Motivation | Monitoring |
|---|---|---|---|
| : Violence, T2 | |||
| 1: | Pushed and ignored | None | |
| 2: | Pushed and ignored | Police officers | |
| 3: | Pushed twice, iPod broken | None | |
| 4: | Pushed twice, iPod broken | Police officers |
## mean sd 5.5% 94.5% n_eff Rhat
## a[1] -0.74 0.19 -1.05 -0.43 2913.63 1
## a[2] -0.82 0.20 -1.16 -0.50 2547.06 1
## a[3] 0.68 0.18 0.40 0.97 2591.94 1
## a[4] -0.18 0.20 -0.49 0.14 2983.36 1
## b[1] 1.50 0.39 0.89 2.14 2884.31 1
## b[2] 0.98 0.47 0.22 1.71 2698.71 1
## b[3] 1.23 0.42 0.56 1.91 3015.93 1
## b[4] 1.16 0.49 0.40 1.94 2188.11 1Comparison: violence, T1 vs T2.
Results show that adolescents, in general, are unlikely to choose violence as a response to a provocation at time 1, but somewhat more likely at time 2. Across both time points, and across scenarios, more crime-prone adolescents are more likely to choose violence as projected act. Adolescents under scenarios involving a higher level of friction (an insult at time 1, and a second push at time 2) are more likely to report projected violence. Level of monitoring seems to have a larger effect at time 2 than at time 1.
The results speak in favour of the perception-choice process, as the perception of a violent act as a viable action alternative seems to be triggered by a higher level of friction (provocation), and subsequently affected by monitoring (influencing the choice-process).
As a final detail, below you find traceplots for each model (theft: t1 and t2; violence: t1 and t2). They all indicate healthy simulations.
traceplot(mst1) #Theft, time 1
traceplot(mst2) #Theft, time 2
traceplot(vt1) #Theft, time 1
traceplot(vt2) #Theft, time 2
